Theorem rpnnen1lem5 12368

Description: Lemma for rpnnen1 12370. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
rpnnen1lem.1	⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
rpnnen1lem.2	⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
rpnnen1lem.n	⊢ ℕ ∈ V
rpnnen1lem.q	⊢ ℚ ∈ V

Assertion

Ref	Expression
rpnnen1lem5	⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥)

Distinct variable groups:   𝑘,𝐹,𝑛,𝑥   𝑇,𝑛

Allowed substitution hints:   𝑇(𝑥,𝑘)

Proof of Theorem rpnnen1lem5

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpnnen1lem.1	. . . 4 ⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}
2		rpnnen1lem.2	. . . 4 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
3		rpnnen1lem.n	. . . 4 ⊢ ℕ ∈ V
4		rpnnen1lem.q	. . . 4 ⊢ ℚ ∈ V
5	1, 2, 3, 4	rpnnen1lem3 12366	. . 3 ⊢ (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥)
6	1, 2, 3, 4	rpnnen1lem1 12365	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) ∈ (ℚ ↑m ℕ))
7	4, 3	elmap 8418	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (ℚ ↑m ℕ) ↔ (𝐹‘𝑥):ℕ⟶ℚ)
8	6, 7	sylib 221	. . . . 5 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥):ℕ⟶ℚ)
9		frn 6493	. . . . . 6 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → ran (𝐹‘𝑥) ⊆ ℚ)
10		qssre 12346	. . . . . 6 ⊢ ℚ ⊆ ℝ
11	9, 10	sstrdi 3927	. . . . 5 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → ran (𝐹‘𝑥) ⊆ ℝ)
12	8, 11	syl 17	. . . 4 ⊢ (𝑥 ∈ ℝ → ran (𝐹‘𝑥) ⊆ ℝ)
13		1nn 11636	. . . . . . . 8 ⊢ 1 ∈ ℕ
14	13	ne0ii 4253	. . . . . . 7 ⊢ ℕ ≠ ∅
15		fdm 6495	. . . . . . . 8 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → dom (𝐹‘𝑥) = ℕ)
16	15	neeq1d 3046	. . . . . . 7 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → (dom (𝐹‘𝑥) ≠ ∅ ↔ ℕ ≠ ∅))
17	14, 16	mpbiri 261	. . . . . 6 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → dom (𝐹‘𝑥) ≠ ∅)
18		dm0rn0 5759	. . . . . . 7 ⊢ (dom (𝐹‘𝑥) = ∅ ↔ ran (𝐹‘𝑥) = ∅)
19	18	necon3bii 3039	. . . . . 6 ⊢ (dom (𝐹‘𝑥) ≠ ∅ ↔ ran (𝐹‘𝑥) ≠ ∅)
20	17, 19	sylib 221	. . . . 5 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → ran (𝐹‘𝑥) ≠ ∅)
21	8, 20	syl 17	. . . 4 ⊢ (𝑥 ∈ ℝ → ran (𝐹‘𝑥) ≠ ∅)
22		breq2 5034	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑛 ≤ 𝑦 ↔ 𝑛 ≤ 𝑥))
23	22	ralbidv 3162	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦 ↔ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥))
24	23	rspcev 3571	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦)
25	5, 24	mpdan 686	. . . 4 ⊢ (𝑥 ∈ ℝ → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦)
26		id 22	. . . 4 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ)
27		suprleub 11594	. . . 4 ⊢ (((ran (𝐹‘𝑥) ⊆ ℝ ∧ ran (𝐹‘𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦) ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹‘𝑥), ℝ, < ) ≤ 𝑥 ↔ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥))
28	12, 21, 25, 26, 27	syl31anc 1370	. . 3 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) ≤ 𝑥 ↔ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥))
29	5, 28	mpbird 260	. 2 ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) ≤ 𝑥)
30	1, 2, 3, 4	rpnnen1lem4 12367	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ)
31		resubcl 10939	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ) → (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ∈ ℝ)
32	30, 31	mpdan 686	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ∈ ℝ)
33	32	adantr 484	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ∈ ℝ)
34		posdif 11122	. . . . . . . . . 10 ⊢ ((sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 ↔ 0 < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
35	30, 34	mpancom 687	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 ↔ 0 < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
36	35	biimpa 480	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → 0 < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )))
37	36	gt0ne0d 11193	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ≠ 0)
38	33, 37	rereccld 11456	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) ∈ ℝ)
39		arch 11882	. . . . . 6 ⊢ ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) ∈ ℝ → ∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘)
40	38, 39	syl 17	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → ∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘)
41	40	ex 416	. . . 4 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → ∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘))
42	1, 2	rpnnen1lem2 12364	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℤ)
43	42	zred 12075	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℝ)
44	43	3adant3 1129	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘) → sup(𝑇, ℝ, < ) ∈ ℝ)
45	44	ltp1d 11559	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘) → sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1))
46	33, 36	jca 515	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) → ((𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ∈ ℝ ∧ 0 < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
47		nnre 11632	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
48		nngt0 11656	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → 0 < 𝑘)
49	47, 48	jca 515	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
50		ltrec1 11516	. . . . . . . . . . . . 13 ⊢ ((((𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) ∈ ℝ ∧ 0 < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 ↔ (1 / 𝑘) < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
51	46, 49, 50	syl2an 598	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 ↔ (1 / 𝑘) < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
52	30	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ)
53		nnrecre 11667	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
54	53	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
55		simpll 766	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
56	52, 54, 55	ltaddsub2d 11230	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 ↔ (1 / 𝑘) < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))))
57	12	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ran (𝐹‘𝑥) ⊆ ℝ)
58		ffn 6487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑥):ℕ⟶ℚ → (𝐹‘𝑥) Fn ℕ)
59	8, 58	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) Fn ℕ)
60		fnfvelrn 6825	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑥) Fn ℕ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥)‘𝑘) ∈ ran (𝐹‘𝑥))
61	59, 60	sylan 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥)‘𝑘) ∈ ran (𝐹‘𝑥))
62	57, 61	sseldd 3916	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥)‘𝑘) ∈ ℝ)
63	30	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ)
64	53	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
65	12, 21, 25	3jca 1125	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → (ran (𝐹‘𝑥) ⊆ ℝ ∧ ran (𝐹‘𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦))
66	65	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (ran (𝐹‘𝑥) ⊆ ℝ ∧ ran (𝐹‘𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦))
67		suprub 11589	. . . . . . . . . . . . . . . . 17 ⊢ (((ran (𝐹‘𝑥) ⊆ ℝ ∧ ran (𝐹‘𝑥) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑦) ∧ ((𝐹‘𝑥)‘𝑘) ∈ ran (𝐹‘𝑥)) → ((𝐹‘𝑥)‘𝑘) ≤ sup(ran (𝐹‘𝑥), ℝ, < ))
68	66, 61, 67	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥)‘𝑘) ≤ sup(ran (𝐹‘𝑥), ℝ, < ))
69	62, 63, 64, 68	leadd1dd 11243	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)))
70	62, 64	readdcld 10659	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ)
71		readdcl 10609	. . . . . . . . . . . . . . . . 17 ⊢ ((sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) → (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ)
72	30, 53, 71	syl2an 598	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ)
73		simpl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
74		lelttr 10720	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ ∧ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) ∧ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥) → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
75	74	expd 419	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ∈ ℝ ∧ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) → ((sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥)))
76	70, 72, 73, 75	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) ≤ (sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) → ((sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥)))
77	69, 76	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
78	77	adantlr 714	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((sup(ran (𝐹‘𝑥), ℝ, < ) + (1 / 𝑘)) < 𝑥 → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
79	56, 78	sylbird 263	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑘) < (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < )) → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
80	51, 79	sylbid 243	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
81	42	peano2zd 12078	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(𝑇, ℝ, < ) + 1) ∈ ℤ)
82		oveq1 7142	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = (sup(𝑇, ℝ, < ) + 1) → (𝑛 / 𝑘) = ((sup(𝑇, ℝ, < ) + 1) / 𝑘))
83	82	breq1d 5040	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = (sup(𝑇, ℝ, < ) + 1) → ((𝑛 / 𝑘) < 𝑥 ↔ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥))
84	83, 1	elrab2 3631	. . . . . . . . . . . . . . . . 17 ⊢ ((sup(𝑇, ℝ, < ) + 1) ∈ 𝑇 ↔ ((sup(𝑇, ℝ, < ) + 1) ∈ ℤ ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥))
85	84	biimpri 231	. . . . . . . . . . . . . . . 16 ⊢ (((sup(𝑇, ℝ, < ) + 1) ∈ ℤ ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇)
86	81, 85	sylan 583	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇)
87		ssrab2 4007	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ⊆ ℤ
88	1, 87	eqsstri 3949	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑇 ⊆ ℤ
89		zssre 11976	. . . . . . . . . . . . . . . . . . 19 ⊢ ℤ ⊆ ℝ
90	88, 89	sstri 3924	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑇 ⊆ ℝ
91	90	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑇 ⊆ ℝ)
92		remulcl 10611	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑘 · 𝑥) ∈ ℝ)
93	92	ancoms 462	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑘 · 𝑥) ∈ ℝ)
94	47, 93	sylan2 595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑘 · 𝑥) ∈ ℝ)
95		btwnz 12072	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 · 𝑥) ∈ ℝ → (∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥) ∧ ∃𝑛 ∈ ℤ (𝑘 · 𝑥) < 𝑛))
96	95	simpld 498	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 · 𝑥) ∈ ℝ → ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥))
97	94, 96	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥))
98		zre 11973	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℝ)
99	98	adantl 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℝ)
100		simpll 766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ ℝ)
101	49	ad2antlr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
102		ltdivmul 11504	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → ((𝑛 / 𝑘) < 𝑥 ↔ 𝑛 < (𝑘 · 𝑥)))
103	99, 100, 101, 102	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → ((𝑛 / 𝑘) < 𝑥 ↔ 𝑛 < (𝑘 · 𝑥)))
104	103	rexbidva 3255	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥 ↔ ∃𝑛 ∈ ℤ 𝑛 < (𝑘 · 𝑥)))
105	97, 104	mpbird 260	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥)
106		rabn0 4293	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅ ↔ ∃𝑛 ∈ ℤ (𝑛 / 𝑘) < 𝑥)
107	105, 106	sylibr 237	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅)
108	1	neeq1i 3051	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 ≠ ∅ ↔ {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥} ≠ ∅)
109	107, 108	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 𝑇 ≠ ∅)
110	1	rabeq2i 3435	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ 𝑇 ↔ (𝑛 ∈ ℤ ∧ (𝑛 / 𝑘) < 𝑥))
111	47	ad2antlr 726	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → 𝑘 ∈ ℝ)
112	111, 100, 92	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑘 · 𝑥) ∈ ℝ)
113		ltle 10718	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ ℝ ∧ (𝑘 · 𝑥) ∈ ℝ) → (𝑛 < (𝑘 · 𝑥) → 𝑛 ≤ (𝑘 · 𝑥)))
114	99, 112, 113	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → (𝑛 < (𝑘 · 𝑥) → 𝑛 ≤ (𝑘 · 𝑥)))
115	103, 114	sylbid 243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ ℤ) → ((𝑛 / 𝑘) < 𝑥 → 𝑛 ≤ (𝑘 · 𝑥)))
116	115	impr 458	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ ℤ ∧ (𝑛 / 𝑘) < 𝑥)) → 𝑛 ≤ (𝑘 · 𝑥))
117	110, 116	sylan2b 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑇) → 𝑛 ≤ (𝑘 · 𝑥))
118	117	ralrimiva 3149	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∀𝑛 ∈ 𝑇 𝑛 ≤ (𝑘 · 𝑥))
119		breq2 5034	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑘 · 𝑥) → (𝑛 ≤ 𝑦 ↔ 𝑛 ≤ (𝑘 · 𝑥)))
120	119	ralbidv 3162	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑘 · 𝑥) → (∀𝑛 ∈ 𝑇 𝑛 ≤ 𝑦 ↔ ∀𝑛 ∈ 𝑇 𝑛 ≤ (𝑘 · 𝑥)))
121	120	rspcev 3571	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑘 · 𝑥) ∈ ℝ ∧ ∀𝑛 ∈ 𝑇 𝑛 ≤ (𝑘 · 𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑇 𝑛 ≤ 𝑦)
122	94, 118, 121	syl2anc 587	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑇 𝑛 ≤ 𝑦)
123	91, 109, 122	3jca 1125	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑇 𝑛 ≤ 𝑦))
124		suprub 11589	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑇 𝑛 ≤ 𝑦) ∧ (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
125	123, 124	sylan 583	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ (sup(𝑇, ℝ, < ) + 1) ∈ 𝑇) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
126	86, 125	syldan 594	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) ∧ ((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥) → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ))
127	126	ex 416	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥 → (sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < )))
128	42	zcnd 12076	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℂ)
129		1cnd 10625	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
130		nncn 11633	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
131		nnne0 11659	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
132	130, 131	jca 515	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
133	132	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
134		divdir 11312	. . . . . . . . . . . . . . . 16 ⊢ ((sup(𝑇, ℝ, < ) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0)) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
135	128, 129, 133, 134	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
136	3	mptex 6963	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) ∈ V
137	2	fvmpt2 6756	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) ∈ V) → (𝐹‘𝑥) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
138	136, 137	mpan2 690	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))
139	138	fveq1d 6647	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → ((𝐹‘𝑥)‘𝑘) = ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘))
140		ovex 7168	. . . . . . . . . . . . . . . . . 18 ⊢ (sup(𝑇, ℝ, < ) / 𝑘) ∈ V
141		eqid 2798	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)) = (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))
142	141	fvmpt2 6756	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℕ ∧ (sup(𝑇, ℝ, < ) / 𝑘) ∈ V) → ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
143	140, 142	mpan2 690	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘))‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
144	139, 143	sylan9eq 2853	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥)‘𝑘) = (sup(𝑇, ℝ, < ) / 𝑘))
145	144	oveq1d 7150	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) = ((sup(𝑇, ℝ, < ) / 𝑘) + (1 / 𝑘)))
146	135, 145	eqtr4d 2836	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) / 𝑘) = (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)))
147	146	breq1d 5040	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (((sup(𝑇, ℝ, < ) + 1) / 𝑘) < 𝑥 ↔ (((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥))
148	81	zred 12075	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (sup(𝑇, ℝ, < ) + 1) ∈ ℝ)
149	148, 43	lenltd 10775	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((sup(𝑇, ℝ, < ) + 1) ≤ sup(𝑇, ℝ, < ) ↔ ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
150	127, 147, 149	3imtr3d 296	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
151	150	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((((𝐹‘𝑥)‘𝑘) + (1 / 𝑘)) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
152	80, 151	syld 47	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥) ∧ 𝑘 ∈ ℕ) → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
153	152	exp31 423	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → (𝑘 ∈ ℕ → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))))
154	153	com4l 92	. . . . . . . 8 ⊢ (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → (𝑘 ∈ ℕ → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → (𝑥 ∈ ℝ → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))))
155	154	com14 96	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (𝑘 ∈ ℕ → ((1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))))
156	155	3imp 1108	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘) → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → ¬ sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1)))
157	45, 156	mt2d 138	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ ∧ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘) → ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥)
158	157	rexlimdv3a 3245	. . . 4 ⊢ (𝑥 ∈ ℝ → (∃𝑘 ∈ ℕ (1 / (𝑥 − sup(ran (𝐹‘𝑥), ℝ, < ))) < 𝑘 → ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥))
159	41, 158	syld 47	. . 3 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥 → ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥))
160	159	pm2.01d 193	. 2 ⊢ (𝑥 ∈ ℝ → ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥)
161		eqlelt 10717	. . 3 ⊢ ((sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥 ↔ (sup(ran (𝐹‘𝑥), ℝ, < ) ≤ 𝑥 ∧ ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥)))
162	30, 161	mpancom 687	. 2 ⊢ (𝑥 ∈ ℝ → (sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥 ↔ (sup(ran (𝐹‘𝑥), ℝ, < ) ≤ 𝑥 ∧ ¬ sup(ran (𝐹‘𝑥), ℝ, < ) < 𝑥)))
163	29, 160, 162	mpbir2and 712	1 ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243   class class class wbr 5030   ↦ cmpt 5110  dom cdm 5519  ran crn 5520   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ↑_m cmap 8389  supcsup 8888  ℂcc 10524  ℝcr 10525  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531   < clt 10664   ≤ cle 10665   − cmin 10859   / cdiv 11286  ℕcn 11625  ℤcz 11969  ℚcq 12336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-sup 8890  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-n0 11886  df-z 11970  df-q 12337

This theorem is referenced by:  rpnnen1lem6  12369